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First-principles Study of Carbon and Vacancy Structures in
Niobium
Denise C. Ford1, Peter Zapol1, and Lance D. Cooley2
1 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439,
USA
2 Technical Division, Ferm i N ational Accelerator Laboratory, Batavia, Il
USA
Abstract
The interstitial chemical impurities hydrogen, oxygen, nitrogen, and carbon are
important for niobium metal production, and particularly for the optimization of niobium
SRF technology. These atoms are present in refined sheets and can be absorbed into
niobium during processing treatments, resulting in changes to the residual resistance and
the performance of SRF cavities. A first-principles approach is taken to study the
properties of carbon in niobium, and the results are compared and contrasted with the
properties of the other interstitial impurities. The results indicate that C will likely form
precipitates or atmospheres around defects rather than strongly bound complexes with
other impurities. Based on the analysis of carbon and hydrogen near niobium lattice
vacancies and small vacancy chains and clusters, the formation of extended carbon chains
and hydrocarbons is not likely to occur. Association of carbon with hydrogen atoms can,
however, occur through the strain fields created by interstitial binding of the impurity
FERMILAB-PUB-15-008-TD ACCEPTED
Operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.
atoms. Calculated electronic densities of states indicate that interstitial C may have a
similar effect as interstitial O on the superconducting transition temperature of Nb.
1. INTRODUCTION
Pure niobium is a primary material for superconducting technology, including
superconducting radio frequency (SRF) cavities, superconducting electronic devices, and
superconducting magnet strands and cables. Manufacturing activities generally strive to
reduce the content of interstitial impurities H, O, N, and C to low levels because they
degrade the superconducting properties, increase electron scattering, and increase the
hardening rate during metal working. Interstitial oxygen, for example, reduces the
superconducting transition temperature (Tc) by ~0.9 K per at. %, and interstitial N and C
are expected to have similar effects.1 Especially high purity, < 30 ppm by mass of O, N,
and C and < 5 ppm by mass H, is specified for Nb grades used in SRF cavities2 to
achieve high thermal conductivity at ~2 K.
Quite a bit of attention has been paid to the elements O and H in niobium, long
ago due to applications of the metal in corrosive environments and in the presence of
superheated steam3 and more recently because etching and polishing of Nb with aqueous
solutions has been used to remove defects introduced during the fabrication of SRF
cavities.4 Niobium forms a complex layering of oxides, including a diffusion gradient of
interstitial oxygen, when the clean metal is exposed to air or water. The oxide is a good
barrier against hydrogen at room temperature, but the structure and composition of the
oxide layers changes upon heating to temperatures as low as 100 °C due to dissolution of
oxygen into the metal.5-6 Niobium absorbs copious amounts of hydrogen during
2
electropolishing7, in conjunction with the electrochemical stripping of the oxides and the
injection of vacancies to form an abundance of vacancy-hydrogen clusters.8-10 If the
hydrogen is not subsequently removed by annealing in high vacuum, niobium hydride
precipitates can form at low temperature, leading to a sharp decrease of the cavity quality
factor Q at the onset of RF field.11 Even when hydrogen is de-gassed by annealing steps,
small precipitates can still form,12 leading to losses at high RF fields. Previously we
analyzed how, single H atoms may be trapped by single O or N atoms to prevent hydride
phase formation.13
Interstitial N has received vigorous attention very recently, due to the apparent
suppression of residual resistance at low RF field and the appearance of a field-dependent
BCS residual resistance component.14 The new behavior results in an increase of the
cavity Q up to moderate fields, and it implies a factor-of-2 reduction in cryogenic
operation cost. Since the introduction of N at high temperature is followed by a brief
electropolishing to remove traces of surface nitrides, it might be possible that the
remaining solid solution of N in Nb completely immobilizes H impurities, N being more
effective than O at trapping H, although this would not explain the unusual physics being
reported. Two theoretical models have been presented15-16 to explain the unusual surface
resistance, where16 pointed out that the addition of N may be crucial for changing Nb
from clean-limit to dirty-limit regimes of BCS residual resistance in the presence of
current flow. Besides nitrogen, titanium, which substitutes for Nb lattice sites and does
not occupy interstitial lattice sites, has been shown to yield the new SRF cavity
behavior.17
Although interstitial carbon in niobium has some similar properties to interstitial
3
oxygen and nitrogen,13 it has not been studied as intensively, perhaps because it has a
much lower solubility.18-19 The lack of attention paid to carbon in niobium may be an
important oversight, however. Recent secondary ion mass spectroscopy measurements of
SRF niobium samples indicate local carbon concentrations of 10-1-10-2 at.% depending on
the specimen history and heat treatment,17 and recent Raman spectroscopy studies
indicate that large amounts of carbon are present in various forms as local clusters on
niobium cavity and sample surfaces.20 How the carbon collects is not known; the reports
above emphasize clean handling and rule out casual environmental contributions.
Dislocations seem to play a role, since material working coupled with chemical polishing
and annealing is a reproducible pathway toward the results above. Some carbide
precipitates exhibit enhanced superconducting gaps, and Raman signatures suggested
extremely strong electron-phonon coupling.21 How different carbon configurations affect
electronic and phonon properties is of interest to understand superconducting properties
of technological Nb materials.
The superconducting properties of bulk niobium carbides and carbo-nitrides have
been studied extensively in the past, as they present an interesting theoretical case study
of electron-phonon superconductivity that is strongly dependent on composition and
structure. The Tc of NbC has been shown to decrease with increasing concentration of
carbon vacancies22 and increase with the substitution of carbon for nitrogen.23-24 NbN
and NbC are very difficult to make at the stoichiometric ratio, which might reflect on
structural instabilities at the root of superconductive electron pairing. Additionally,
NbC1-x up to x ≈ 0.25 with disordered carbon vacancies has been shown to have a lower
Tc than the ordered compound at a comparable concentration.25
4
In addition to carbon or carbide clusters in Nb and the bulk niobium carbide
phases, the interaction of C with Nb is itself an interesting physical and metallurgical
question. NbC precipitates are tremendously important to high-strength corrosion-
resistant steels, because the compound remains stable in austenitic stainless steels at high
temperatures and the strain fields associated with NbC nanoprecipitates act as effective
dislocation pinners.26-27
In the Nb-C system, C atoms might not necessarily precipitate as intermetallic
phases or remain dissolved as a solid solution, but instead they can form atmospheres
around lattice defects such as metal lattice vacancies, dislocations, or grain boundaries. C
may be unique among the interstitial impurities by having a tendency to form dimers or
chains, as C-C dimers in metal vacancies have been predicted for bcc Fe28-30. The
formation of carbon clusters around dislocations has been observed as early as the
1960’s,31-32 and the migration of carbon atoms to grain boundaries upon heat treatment
has recently been observed in SRF niobium.33 Study of the interactions of carbon
impurities with other defects can have a major effect on understanding the kinetic
mechanisms in Nb processing.
In this study, we examine potential structures of carbon impurity and metal
vacancy complexes in bcc niobium. Density functional theory (DFT) is used to compute
the total energies and electronic structures of carbon atoms in bcc niobium interstitial
sites, carbon atoms near niobium lattice vacancies, di-vacancies, and tri-vacancies, and
several niobium carbides. The C-H interaction in niobium is also investigated. The
results are discussed in terms of thermodynamic stability and electronic properties.
5
2. METHODS
Most of the impurity structures were modeled in a 4x4x4 unit cell of bcc niobium
(128 atoms) with Nb atoms removed to create lattice vacancies, and C atoms added in
interstitial or vacancy binding sites. Some of the larger defect configurations required the
use of 5x5x5 unit cells. The unit cells were chosen to be computationally efficient while
minimizing interactions between periodic images. They are not intended to represent a
specific concentration of defects. Several tests were performed for larger unit cells and
the conclusions drawn in this paper were upheld. Nb2C was modeled in the β-LT
structure, which has hexagonal symmetry and contains four formula units per unit cell,
and NbC was modeled in the rock-salt structure with four formula units per unit cell.
All calculations were performed with the Vienna Ab initio Simulation Package
(VASP),34-35 using DFT, periodic boundary conditions, and a plane wave basis set with a
400 eV kinetic energy cutoff. The generalized gradient approximation (GGA) was used
with the Perdew, Burke, Ernzerhof (PBE) exchange–correlation functional,36 and the core
electrons were described by the projector-augmented-wave (PAW) pseudopotentials.37-38
All of the structure optimizations were calculated with all of the atom and cell degrees of
freedom relaxed. Forces were converged to 0.02 eV/A. K-points were selected by the
Monkhorst–Pack39 scheme with 2x2x2 gamma-centered grids for the geometry
optimizations of the defect structures in niobium, 3x10x6 gamma-centered for Nb2C, and
9x9x9 for NbC. Denser grids were used for the electronic densities of states (DOS)
calculations: 6x6x6 gamma-centered grids for Nb defects, 5x19x12 gamma-centered for
Nb2C, and 13x13x13 for NbC. The partial occupancies for the one-electron levels were
6
determined by the first order Methfessel–Paxton method with a smearing width of 0.2 eV
for the geometry optimizations, while the tetrahedron method with Blöchl corrections
was used for the DOS calculations. Spin-polarization was not used for the calculations,
after check for selected configurations resulted in zero magnetic moment. The optimized
lattice parameters for Nb, Nb2C and NbC are within 1 % of their experimentally
determined values.40-42
Several energetic properties are discussed in the results section for comparison
between structures. Binding energies were calculated from various reference states as
were appropriate for the discussion, and these are specified in the results section.
Negative (−) binding energies indicate that the bound state is lower in energy than the
unbound state or reference state. Strain energies were calculated by subtracting the
energy of the niobium lattice that was deformed from defects or impurities from the
energy of the ideal bcc niobium lattice. In the cases of impurity absorption, the geometry
was optimized with the impurity, then the impurity was removed and a single-point
energy calculation was done to determine the energy of the deformed lattice.
3. RESULTS AND DISCUSSION
Carbon impurity atoms occupy octahedral interstitial sites in bcc niobium. The
binding configuration for a single carbon atom in Nb is shown in Figure 1a. As
calculated previously in ref13, absorption of carbon into the octahedral site imparts a large
strain on the niobium lattice substantiated by a strain energy of 0.96 eV, a 25% expansion
of the two “short” C-Nb bonds compared to what they would be in an undeformed
octahedral site, and a 5% contraction of the four “long” C-Nb bonds. This indicates that
7
C should preferentially bind to sites near structural lattice defects such as vacancies,
dislocations, grain boundaries, interfaces, and surfaces; where the lattice strain imparted
by the C atom can be reduced.
Figure 1. Configurations of one carbon atom in (a) an octahedral lattice interstice in bcc
niobium, (b) an interior vacancy binding site, and (c) an exterior vacancy binding site.
The insets show the configurations in a larger section of the Nb lattice. Niobium atoms
are represented as blue spheres, carbon atoms as small black spheres, and niobium lattice
vacancies as grey cubes. Black arrows indicate the direction of relaxation of Nb atoms
upon absorption of a C atom into the binding site.
3.1 Interstitial Impurity Atoms Near Single Lattice Vacancies
The calculated vacancy formation energy for niobium is 2.71 eV, which is in
good agreement with the experimentally determined range 2.6–3.1 eV given in a review
by Schultz and Ehrhart43 and other first-principles calculations44-45. We explored several
binding sites for a C atom near a single niobium lattice vacancy and show two
configurations in Figure 1: the site preferred by H and O impurity atoms (ref 13 and
references therein) is depicted in Figure 1b and the site that we found to be the most
energetically favorable for C is depicted in Figure 1c. The binding energies of C to the
8
vacancy for these two configurations relative to C in an octahedral site far from a
vacancy are 0.00 eV and -0.39 eV, respectively (Table 1). The niobium lattice strain
energies for these two vacancy-binding configurations are 0.30 eV and 0.68 eV,
respectively, which compares to the strain energy of 0.96 eV for C binding in an
octahedral site far away from the vacancy. Although the reduction in strain energy is
larger for C to migrate to a site where one of the short Nb-C bonds is eliminated (Figure
1b), C exhibits a strong preference for octahedral binding and maximizing the electronic
interaction with Nb. This may be understood considering the elemental data provided in
Table 1. The groundstate electron configuration of the C atom has four empty 2p states,
resulting in the ability to form up to four covalent bonds in molecules. Carbon, however,
does not form covalent bonds with Nb atoms in the bcc lattice, as can be seen in the
charge density difference plots in Figure 2. Ionic nature of bonding is observed, as
indicated by the large region of electron density around the C atom, small regions of
decreased electron density between the C and Nb atoms and increased electron density
near the Nb atoms in the opposite direction of the C atom. Carbon has a higher
electronegativity than Nb, so it draws some of the Nb electrons onto itself, resulting in a
partially anionic state. The charge on the carbon atom in three binding configurations
shown in Figure 1 was calculated with the Bader partitioning scheme46 and is given in
Table 1. The most favorable configuration for carbon near a Nb lattice vacancy allows
for a greater charge on the carbon atom than does the less favorable configuration, even
though the less favorable configuration provides more physical space for the C atom to
occupy.
Commented [DF1]: New discussion.
9
Table 1. Elemental data for Nb, C, N, O, and H; interstitial site preference, charge on the
interstitial atom in the Nb lattice, and binding energy (B.E.) for a C, N, O, and H atom in
a Nb lattice vacancy referred to the binding energy for that atom in its preferred
interstitial binding site. Vacancy site b. and vacancy site c. are depicted in Figure 1b and
1c, respectively. The lowest energy is in bold, and n.a. stands for not applicable.
atomic radius (Å)a
Pauling electronegativityb
ground state electronic configuration
interstitial site preferencec
charge on interstitial atom (e-)c
Nb 1.45 1.60 [Kr] 4d4 5s1 n.a. n.a. C 0.70 2.55 [He] 2s2 2p2 octahedral -1.76 N 0.65 3.04 [He] 2s2 2p3 octahedral -1.63 O 0.60 3.44 [He] 2s2 2p4 octahedral -1.35 H 0.25 2.20 1s1 tetrahedral -0.65
B.E. (eV) for vacancy site b.
charge on interstitial atom (e-)
B.E. (eV) for vacancy site c.
charge on interstitial atom (e-)
C 0.00 -1.60 -0.39 -1.75 N -0.25 -1.55 -0.29 -1.64 O -0.79 -1.32 -0.22 -1.37 H -0.32 -0.56 n.a. n.a.
a Reference 47
b Reference 48-49
c Reference 13
Commented [DF2]: Moved; used to appear after 2 carbons near a vacancy was discussed. Charges on the interstitial impurity atoms were also added.
10
Figure 2. Charge density difference plots referred to the deformed niobium lattice with
carbon atom(s) removed corresponding to the configurations in Figure 1a (a), Figure 1c
(b), Figure 3a (c), and Figure3b (d). The isosurface levels are 0.027 e-/Å3. The charge
density difference referred to the overlapping atomic density is also shown for the
configurations in (c) and (d), with isosurface levels at 0.101 e-/Å3. Niobium atoms are
represented as light blue spheres and carbon atoms as small black spheres. The blue
regions of charge density represent charge depletion and the red regions represent charge
enhancement.
The basic elemental data, interstitial site preferences, vacancy binding energies,
and charges in the bcc niobium lattice for a single H, O, and N atom are also reported in
Commented [DF3]: New figure.
11
Table 1. Both interstitial and vacancy site preferences are correlated with the ground
state electronic configuration of the impurity atom and not with its size. H is s-shell and
prefers the sites in the Nb lattice with lower electron density (e.g. tetrahedral as opposed
to octahedral interstitial sites). The other impurities are p-shell and prefer the octahedral
interstitial site. Their preference for binding near Nb vacancies – Figure 1b or 1c –
depends on the number of 2p electrons in their ground state electron configuration: more
electrons in the p-shell indicates a stronger preference for the configuration depicted in
Figure 1b. H, O, and N are also more electronegative than Nb, therefore become partially
anionic in the bcc niobium lattice, as discussed above for C. The ability of each impurity
atom to maintain its charge when binding near a vacancy correlates with its vacancy site
preference. For example, O is able to maintain approximately the same charge in either
of the vacancy binding sites considered as it had in an octahedral site far from the
vacancy, therefore will bind in an interior vacancy site (Figure 1b) to minimize the lattice
strain. C, on the other hand, looses 0.16 e- when binding in an interior binding site
compared to an octahedral site far from the vacancy, whereas it maintains the same
charge when binding in an exterior binding site close to the vacancy. N represents an
intermediate case. The magnitude of charge on the impurity atoms is ordered as
C>N>O>H, which follows the order of the atom sizes and the number of empty states in
the atom’s ground state electron configuration rather than the electronegativity.
Figure 3 shows three possible configurations of two carbon atoms near a niobium
lattice vacancy. We calculate an binding energy of -0.14 eV for two C atoms in interior
octahedral binding sites that are directly across from each other (Figure 3a) relative to C
atoms in octahedral interstitial sites far away from each other and any lattice vacancies.
Commented [DF4]: Moved and modified according to new location and discussion; this paragraph used to be the last paragraph in this section.
12
A C-C dimer with a bond length of 1.45 Å, which is in between the C-C single and
double bond lengths, and a binding energy of -0.46 eV forms when two C atoms were
placed in neighboring vacancy binding sites (Figure 3b). The electron density
distributions for these two configurations are shown in Figure 2c and 2d, respectively.
Small regions of reduced electron density between the Nb and C atoms grew when the
isolevels were decreased below what is shown in the figure, indicating that no covalent
bonding occurs between the Nb and C atoms. Although it appears that the C atoms as
shown in the first panel in Figure 2c may contain some overlapping electron density, no
indication of covalent bonding was found between any of the atoms in this configuration
when the overlapping atomic densities is taken as the reference (second panel). When
the overlapping atomic densities is taken as the reference for the configuration appearing
to contain a C-C dimer, an increased electron density between the two C atoms is
apparent (second panel of Figure 2d), depicting the covalent nature of their bonding. The
regions of reduced electron density surrounding the Nb atoms in both configurations,
when the overlapping atomic densities is taken as the reference, indicates the metallic
nature of the Nb bonding. The energetically most favorable configuration for two C
atoms near a vacancy is neither of these configurations, however; it is instead where each
C atom maintains an octahedral configuration within the Nb lattice and one of the short
Nb-C bonds is with a Nb atom at the corner of the cube surrounding the vacancy (Figure
3c). This configuration allow the C atoms to maintain the maximum electronic
interaction with the Nb atoms, while reducing lattice strain, as discussed above for the
configurations of single carbon atoms near single lattice vacancies. The binding energy
for this configuration is -0.89 eV. Note, this configuration is preferred to configurations
Commented [DF5]: New discussion.
13
where the C atoms are in octahedral binding sites near the vacancy and the short Nb-C
bonds push the Nb atoms at the corners of the vacancy cube towards each other, which
would cause repulsion between the Nb atoms; or both C atoms are bonded to the same
vacancy-corner Nb, which would not provide sufficient electrons to the C atoms. The
ability of niobium to accommodate C in vacancy binding sites as well as in the strain
field around the vacancy suggests the possibility of Cottrell atmospheres50 around these
defect sites.
Figure 3. Configurations of two carbon atoms near a niobium lattice vacancy where they
are in interior vacancy binding sites directly across from each other (a), offset from
neighboring interior vacancy binding sites such that they form a dimer (b) and in the
lowest energy exterior binding site configuration (c). Niobium atoms are represented as
blue spheres, carbon atoms as small black spheres, and niobium lattice vacancies as grey
cubes.
Carbon is unique among the interstitial impurity atoms – H, O, N, and C – in that
it can form dimers in Nb lattice defects. We attempted to place pairs of interstitial atoms
– H-H, O-O, N-N, and C-C – with gas phase dimer bond lengths in configurations similar
to Figure 3b in a Nb lattice vacancy, as well as with one interstitial atom in an interior
a. b. c.
Commented [DF6]: Modified to emphasize the electronic interaction.
14
vacancy binding site and the other pointed towards the center of the vacancy. All pairs
except for C-C dissociated. This may be rationalized by considering the basic elemental
data shown in Table 1. Carbon can form up to four covalent bonds to fill its 2p orbitals,
so if it is in a position in the Nb lattice where it is facing a lower electron density, such as
next to a Nb vacancy, it will form a covalent bond with an available neighboring carbon
atom. The other p-block interstitial atoms considered here have more electrons in their
2p orbitals, therefore requiring less bonding electrons from their neighbors. C-C single
bonds are stronger than N-N and O-O single bonds. Additionally, N and O have higher
electronegativities than carbon, enabling them to more easily extract electrons from each
neighboring Nb atom. H also has a higher electronegativity than Nb, therefore preferring
to bond with Nb over another H atom.
3.2 Niobium Vacancy Dimer and Trimer Configurations and Their Interaction with
Carbon Impurity Atoms
Optimized geometries for calculated configurations of niobium dimer and trimer
vacancy clusters are shown in Figure 4. We find the second-nearest neighbor di-vacancy
to be the most stable di-vacancy configuration, in agreement with previous
experimental51 and computational52 studies, with a binding energy of -0.38 eV. The
binding energy for vacancy trimers is approximately equal to the sum of the binding
energies of vacancy pairs, and a triangle with two sides formed from pairs of nearest-
neighbor di-vacancies and the third side formed from a pair of second-nearest neighbor
di-vacancies is the most stable tri-vacancy configuration. Vacancies tend to migrate to
sinks, such as surfaces, when they are mobile; however if they find another vacancy or
15
vacancy cluster first, they may become trapped. Reported vacancy migration energies
range from 0.6-1.0 eV,43, 53 which is less than the 1.36-1.44 eV activation energy reported
for C diffusion between octahedral interstitial sites.54-55 Therefore, it is possible to form
small vacancy clusters prior to vacancy trapping by C atoms. This conclusion would still
apply in the cases where interstitial O, with an activation energy for diffusion of 1.17
eV,55 or interstitial N, with an activation energy of 1.52 eV,55 are present.
Figure 4. Configurations of niobium lattice defects and their interaction energies (in eV)
referred to single isolated vacancies. The formation energy of a single vacancy is 2.71
eV, and the lowest interaction energies for di- and tri- vacancies are in bold. Niobium
atoms are represented as blue spheres and niobium lattice vacancies as grey cubes.
Additionally, vacancy dimers and linear trimers are reminiscent of edge
dislocation cores, with the nearest-neighbor pair (chain) being in the [111] direction, the
second-nearest neighbor pair (chain) being in the [100] direction, the third-nearest
neighbor pair (chain) being in the [110] direction, and the fourth-nearest neighbor pair
(chain) being in the [311] direction. Since the niobium atoms surrounding a vacancy
relax towards the center of the vacancy, their Nb-Nb distances are shorter than in the
16
bulk, so we speculate that the strain-field around a vacancy dimer (linear trimer) may be
more representative of the compressive side of an edge dislocation than the expansive
side. The non-linear trimer configurations are similarly related to the lattice planes, and
therefore planar defects.
We calculated several binding configurations of carbon near first- and second-
nearest neighbor di-vacancies and again found that configurations in which the carbon
atom maintains full octahedral coordination with niobium atoms are more stable than
configurations in which one of the niobium atoms is replaced by a vacancy. In some
cases the binding energy for a C atom near a di-vacancy was more favorable than for a C
atom near a single vacancy because there is more space in the di-vacancy configuration
for the Nb atom forming one of the short Nb-C bonds to be displaced. Examples of such
a configuration for a first- and a second-nearest neighbor di-vacancy are shown in Figure
5. Both of these configurations have a binding energy of -0.49 eV, referred to a C atom
in an octahedral site far away from the di-vacancy. Two configurations of C near a Nb
tri-vacancy cluster are also shown in Figure 5. The configuration of C near the vacancy
cluster in the (110) plane has a lower binding energy (-0.72 eV) than C near a single
vacancy, whereas C near a (100) vacancy cluster has a comparable binding energy (-0.39
eV). The enthalpy of segregation to grain boundaries for C in Nb has been determined
from the radio-tracer serial sectioning technique to be 49 kJ/mol,56 which is in-line with
our calculation of multiple potential configurations of C near planar defects with binding
energies of that magnitude. Bokstein and Razumovskii56 also measured C diffusion in
niobium at 800–1173 K and report a lower activation energy and higher prefactor for
grain boundary diffusion compared with bulk diffusion, which corroborates with our
17
finding of weaker binding energies for the configurations with less C-Nb bonds in
extended lattice defects than for the configurations with octahedrally coordinated C in the
strain field around the lattice defects. Carbon diffusion is accelerated by ~2 orders of
magnitude in grain boundaries.56-57
Figure 5. Configurations of a C atom near Nb di- and tri-vacancy clusters that have a
lower binding energy than a C atom near a single Nb vacancy (a) and a configuration of
C near a Nb tri-vacancy that has a higher binding energy than a C near a single Nb
vacancy (b). Niobium atoms are represented as blue spheres, carbon atoms as black
spheres, and niobium lattice vacancies as grey cubes.
C-C dimers are also more stable inside of a Nb nearest-neighbor di-vacancy than
inside of a single vacancy. The configuration shown in Figure 6a, for example, has a
binding energy of -0.60 eV, referred to isolated interstitial C atoms far from the di-
vacancy. The binding energy reduces to -0.23 eV, however, when a trimer is formed
(Figure 6b). Although we do not expect C chains to form in Nb defects, because the C-C
dimer bond length is already longer than a double bond and the binding energy of a
trimer in a di-vacancy is already reduced compared to a the binding energy of a dimer,
recent Raman spectroscopy measurements20 indicate that we should consider the
possibility more carefully. Therefore, we also selected two vacancy trimer configurations Commented [DF7]: New sentence about bonding.
18
– the nearest-neighbor linear trimer and the lowest energy trimer cluster – to examine to
possibility of forming chains of C atoms in extended Nb lattice defects. We found that a
C3 chain inside of a vacancy chain is less stable than C atoms in nearby octahedral sites
around the vacancy chain (similar to Figure 1c). Inside of the vacancy cluster,
configurations of a C-C dimer and a separate C were also more stable than the C3 chain.
Figure 6. Configurations of a C-C dimer (a) and C3 chain (b) near a nearest neighbor Nb
di-vacancy. Niobium atoms are represented as blue spheres, carbon atoms as small black
spheres, and niobium lattice vacancies as grey cubes.
3.3 Comparison Among Niobium Carbide Phases
The binding energies of selected configurations of carbon in niobium as well as
the ordered niobium carbide phases are shown in Figure 7. Although the configurations
of carbon near niobium lattice defects are favorable compared to interstitial carbon far
away from lattice defects, Nb2C has the lowest energy so will preferentially form under
favorable kinetic conditions. Nb2C has been formed in sputter-deposited thin films with
compositions as low as 1.9% C.19 Carbon has been shown to migrate to dislocations and
replace oxygen in that region during strain aging experiments at 155 °C,32 which may
convert to carbide phases once a sufficient local concentration has been reached.
19
Additional ordered phases that are not included in this study are present in experimental
phase diagram18 and others have been predicted by first principles calculations58 as well.
Figure 7. Binding energies of selected configurations of C in Nb relative to isolated C
atoms in octahedral interstitial sites and niobium lattice vacancies or nearest-neighbor
(NN) di-vacancies.
3.4 Density of States and Trends in Superconductivity for NbCx Systems
The C and Nb partial electronic DOS for pure Nb, interstitial C in Nb, Nb2C, and
NbC are shown in Figure 8 and corresponding DOS at the Fermi level (NF) for these and
selected other configurations of carbon in niobium are given in Table 2. The metallic
band of pure bcc Nb in Figure 8 is ~9 eV wide with a peak at the Fermi level. As C is
added to the octahedral interstitial sites in Nb, C 2p states appear around -5 eV and bind
with Nb 5s electrons; this lowers NF. Although NF is not clearly correlated with Tc
20
because it has both a direct proportionality and an indirect effect on the log-mean phonon
frequency,59 Koch, et al.60 has studied the Nb - interstitial O system with a variety of
experimental techniques and showed that the superconducting transition temperature, NF,
and electron-phonon coupling constant decrease with increasing oxygen concentration.
Additionally, Dynes and Varma analyzed data for a number of transition metal alloys,
including Nb-O, and concluded that the electron-phonon coupling parameter varies
linearly with NF.61 We calculate a similar drop-off of NF for interstitial C in Nb as Koch,
et al. report for interstitial O in Nb, and show in ref13 that interstitial C deforms the Nb
lattice is a similar way as O, but to a larger extent. Therefore, we predict a similar effect
of interstitial C on niobium superconductivity as interstitial O imparts. Collection of
carbon atoms around niobium vacancy-type defects; however, mitigates this effect (Table
2), as the portion of the effect due to lattice deformation is reduced.
As shown in Figure 8, the electronic structure of the niobium carbide alloy
changes considerably as the carbon content increases and the crystal structure of the Nb
lattice changes from bcc. The rock-salt structure of NbC has a lower NF (Table 2) than
Nb, but a higher Tc. Density functional theory calculations of the Fermi surface of NbC
show that it contains nesting features that enhance the electron-phonon coupling.62 The
DOS for Nb2C is intermediate in appearance between Nb with interstitial C and NbC,
with strong overlap between the Nb and C states below -3 eV, and mostly Nb states
above. NF is lower for Nb2C than for Nb and NbC. Therefore, we suggest that Tc for
Nb2C is likely less than for Nb or NbC.
21
Figure 8. Partial electronic DOS of selected configurations of carbon in niobium, plotted
with the Fermi energy at 0 eV.
22
Table 2. DOS at the Fermi level (NF) for selected configurations of C in Nb.
NF (# / eV-atom) Nb 1.53 Nb with 0.78% vacancies 1.51 Nb with 1.56% vacancies as nearest-neighbor di-vacancies 1.64 Nb with 0.78% C in octahedral sites 1.37 Nb with single C atoms near single vacancies (0.78%) 1.47 Nb with 1.56% C as 2 atoms near single vacancies 1.43 Nb with 1.56% C as dimers near vacancies 1.40 Nb2C 0.21 NbC 0.38
3.5 Interaction Between Carbon and Hydrogen in the Niobium Lattice
Next we calculate formation energies and structures of possible C-H complexes
from C and H impurities in bcc Nb. We considered a C-H dimer inside of a single
niobium vacancy and found that while it is stable, is less favorable by 0.20 eV compared
to isolated interstitial C and H atoms far away from the vacancy. We also considered
simple hydrocarbons C3H8, C3H5, and C3H2 in a chain of second-nearest neighbor Nb
lattice vacancies, and in all cases the H atoms spontaneously dissociated from the C
atoms during geometry optimization.
Interstitial carbon atoms may, however, serve as trapping centers for interstitial
hydrogen atoms. We calculated the binding energy for an H atom in a tetrahedral
interstitial site 2-8.5 Å away from a C atom in an octahedral site and show the results in
Figure 9 compared to the profiles for H trapping by O and N. All three profiles are
similar, and the trapping energy is slightly greater for C and N than for O. The
configuration for the lowest energy C (N, O) – H pair is also shown in Figure 9. The C
23
atom shares one bonding Nb atom with the H atom, and this Nb atom is displaced in the
same direction resulting from its interaction with the C atom and the H atom.
Figure 9. Plot of binding energy, referred to an H atom in a Nb tetrahedral interstitial
site and a C, N, or O atom in an infinitely far away octahedral interstitial site, versus the
distance between a C, N, or O atom in an octahedral site and an H atom in a nearby
tetrahedral site. The O data series was previously published in ref13. The highest-energy
configuration has a distance between the atoms of less than 2.5 Å and is not shown on the
plot. An excerpt of lowest energy configuration is also shown; Nb atoms are represented
as blue spheres, the C, N, or O is a small black sphere and the H is a small dark blue
sphere.
4. Conclusions
First-principles calculations of carbon in Nb and its interactions with Nb vacancy-
type defects, other C atoms, and H atoms show that C is likely to form precipitates or
atmospheres around the vacancy-type defects rather than strongly bound complexes with
24
other impurities. While Nb2C is calculated to be the most stable phase for C in Nb,
carbon-vacancy complexes can form stable states, which are competitive with NbC and
may be favored kinetically in Nb processing. These are likely to be carbon atoms in
octahedral interstitial positions near vacancies or extended lattice defects.
Carbon is unique among the interstitial impurities – H, O, N, and C – in that it can
form dimers inside of Nb lattice vacancies. Extended chains of carbon atoms and
hydrocarbon inclusions, however, don’t seem to be favorable. Association of H with C
atoms does occur in niobium through the strain field created by binding in interstitial
sites, but a C-H bond does not form.
The electronic DOS at the Fermi level decreases for interstitial C, similar to
interstitial O, indicating that a similar decrease in Tc may follow. Binding of C near
lattice defects somewhat mitigates this effect.
Acknowledgements
We acknowledge computer resources from Fermilab, Argonne LCRC, and Argonne
Center for Nanoscale Materials. Argonne National Laboratory, a U.S. Department of
Energy Office of Science laboratory, is operated under Contract No. DE-AC02-
06CH11357.
References
1. DeSorbo, W., Effect of Dissolved Gases on Some Superconducting Properties of Niobium. Phys. Rev. 1963, 132, 107-121. 2. Cooley, L., Technical Specifications for High RRR Grade Niobium Sheet for Use in Superconducting Radio Frequency Cavities Fermilab Specification 5500.000-ES-371037 2011.
25
3. Bauer, A. A.; Berry, W. E.; DeMastry, J. A.; Koester, R. D.; Rough, F. A., Development of Niobium Alloys Resistant to Superheated Steam; Battelle Memorial Institute: Columbus, 1964. 4. Padamsee, H., Rf Superconductivity Science,Technology and Applications; Wiley–VCH: Weinheim, 2009; Vol. 2. 5. Ma, Q.; Ryan, P.; Freeland, J. W.; Rosenberg, R. A., Thermal Effect on the Oxides on Nb(100) Studied by Synchrotron-Radiation X-Ray Photoelectron Spectroscopy. J. Appl. Phys. 2004, 96, 7675-7680. 6. Delheusy, M.; Stierle, A.; Kasper, N.; Kurta, R. P.; Vlad, A.; Dosch, H.; Antoine, C.; Resta, A.; Lundgren, E.; Andersen, J., X-Ray Investigation of Subsurface Interstitial Oxygen at Nb/Oxide Interfaces. Appl. Phys. Lett. 2008, 92, 101911. 7. Ricker, R. E.; Myneni, G. R., Evaluation of the Propensity of Niobium to Absorb Hydrogen During Fabrication of Superconducting Radio Frequency Cavities for Particle Accelerators. J. Res. Natl. Inst. Stand. Technol. 2010, 115, 353-371. 8. Fukai, Y.; Okuma, N., Evidence of Copious Vacancy Formation in Ni and Pd under a High Hydrogen Pressure. Jpn. J. Appl. Phys. 1993, 32, L1256-L1259. 9. Fukai, Y.; Okuma, N., Formation of Superabundant Vacancies in Pd Hydride under High Hydrogen Pressures. Phys. Rev. Lett. 1994, 73, 1640-1643. 10. Koike, H.; Shizuku, Y.; Yazaki, A.; Fukai, Y., Superabundant Vacancy Formation in Nb–H Alloys; Resistometric Studies. J. Phys.: Condens. Matter 2004, 16, 1335–1349. 11. Knobloch, J., The "Q Disease" in Superconducting Niobium Rf Cavities. In First International Workshop on Hydrogen in Materials and Vacuum Systems, Myneni, G. R.; Chattopadhyay, S., Eds. AIP: Melville, New York, 2003; pp 133-150. 12. Ciovati, G.; Myneni, G.; Stevie, F.; Maheshwari, P.; Griffis, D., High Field Q Slope and the Baking Effect: Review of Recent Experimental Results and New Data on Nb Heat Treatments. Phys. Rev. ST Accel. Beams 2010, 13, 022002. 13. Ford, D. C.; Cooley, L. D.; Seidman, D. N., Suppression of Hydride Precipitates in Niobium Superconducting Radio-Frequency Cavities. Supercond. Sci. Technol. 2013, 26, 105003. 14. Grassellino, A.; Romanenko, A.; Sergatskov, D.; Melnychuk, O.; Trenikhina, Y.; Crawford, A.; Rowe, A.; Wong, M.; Khabiboulline, T.; Barkov, F., Nitrogen and Argon Doping of Niobium for Superconducting Radio Frequency Cavities: A Pathway to Highly Efficient Accelerating Structures. Supercond. Sci. Technol. 2013, 26, 102001. 15. Xiao, B. P.; Reece, C. E. A New First-Principles Calculation of Field-Dependent Rf Surface Impedance of Bcs Superconductor and Application to Srf Cavities 2014. 16. Gurevich, A., Reduction of Dissipative Nonlinear Conductivity of Superconductors by Static and Microwave Magnetic Fields. PRL 2014, 113, 087001. 17. Dhakal, P., et al., Effect of High Temperature Heat Treatments on the Quality Factor of a Large-Grain Superconducting Radio-Frequency Niobium Cavity. Phys. Rev. ST Accel. Beams 2013, 16, 042001. 18. Huang, W., C-Nb Phase Diagram; ASM International: Materials Park, OH, 2006; Vol. http://www1.asminternational.org/AsmEnterprise/APD. 19. Volodin, V. N.; Teleushev, Y. Z.; Zhakanbaev, E. A., Structure and Phase Composition of Nb–C Deposited Films. Phys. Met. 2013, 114, 395-399. 20. Cao, C.; Ford, D.; Bishnoi, S.; Proslier, T.; Albee, B.; Hommerding, E.; Korczakowski, A.; Cooley, L.; Ciovati, G.; Zasadzinski, J. F., Detection of Surface
26
Carbon and Hydrocarbons in Hot Spot Regions of Niobium Superconducting Rf Cavities by Raman Spectroscopy. Phys. Rev. ST Accel. Beams 2013, 16, 064701. 21. Cao, C., et al., Giant Two-Phonon Raman Scattering from Nanoscale Nbc Precipitates in Nb Phys. Rev. B, submitted. 22. Giorgi, A. L.; Szklarz, E. G.; Storms, E. K.; Bowman, A. L.; Matthias, B. T., Effect of Composition on the Superconducting Transition Temperature of Tantalum Carbide and Niobium Carbide. Phys. Rev. 1962, 125, 837-838. 23. Toth, L. E., Transition Metal Carbides and Nitrides; Academic Press: New York, 1971. 24. Blackburn, S.; Cote, M.; Louie, S.; Cohen, M. L., Enhanced Electron-Phonon Coupling near the Lattice Instability of Superconducting Nbc1-Xnx from Density-Functional Calculations. Phys. Rev. B 2011, 84, 104506. 25. Rempe, A. I. G. A. A., Superconductivity in Disordered and Ordered Niobium Carbide. Phys. Stat. Sol. (b) 1989, 151, 211-224. 26. Shrestha, S. L.; Xie, K. Y.; Ringer, S. P.; Carpenter, K. R.; Smith, D. R.; Killmorec, C. R.; Cairneya, J. M., The Effect of Clustering on the Mobility of Dislocations During Aging in Nb-Microalloyed Strip Cast Steels: In Situ Heating Tem Observations. Sripta Mater. 2013, 69, 481–484. 27. Dutta, B.; Palmiere, E. J.; Sellars, C. M., Modelling the Kinetics of Strain Induced Precipitation in Nb Microalloyed Steels. Acta Mater. 2001, 49, 785-794. 28. Johnson, R. A.; Dienes, G. J.; Damask, A. C., Calculations of the Energy and Migration Characteristics of Carbon and Nitrogen in a-Iron and Vanadium. Acta Metall. 1964, 12, 1215-1224. 29. Domain, C.; Becquart, C. S.; Foct, J., Ab Initio Study of Foreign Interstitial Atom (C, N) Interactions with Intrinsic Point Defects in a-Fe. Phys. Rev. B 2004, 69, 144112. 30. Fu, C.-C.; Meslin, E.; Barbu, A.; Willaime, F.; Oison, V., Effect of C on Vacancy Migration in a-Iron. Solid State Phenom. 2008, 139, 157-164. 31. Szkopiak, Z. C.; Derry, L. W., Some Strain-Ageing Effects in Commercial Niobium. J. Nucl. Mater. 1964, 13, 130-136. 32. Szkopiak, Z. C.; Miodownik, A. P., The Mechanism of Strain-Ageing in Commercially Pure Niobium. J. Nucl. Mater. 1965, 17, 20-29. 33. Maheshwari, P.; Zhou, C.; Stevie, F. A.; Myneni, G. R.; Spradlin, J.; Ciovati, G.; Rigsbee, J. M.; Batchelor, A. D.; Griffis, D. P., Sims and Tem Analysis of Niobium Bicrystals. In Proceedings of SRF2011, Chicago, IL, 2011; p THPO028. 34. Kresse, G.; Furthmuller, J., Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169-11186. 35. Kresse, G.; Hafner, J., Abinitio Molecular-Dynamics for Liquid-Metals. Phys. Rev. B 1993, 47, 558-561. 36. Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. 37. Kresse, G.; Joubert, D., From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758-1775. 38. Blochl, P. E., Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953-17979. 39. Monkhorst, H. J.; Pack, J. D., Special Points for Brillouin-Zone Integrations. Phys. Rev. B 1976, 13, 5188-5192.
27
40. Huber, K. P.; Herzberg, G., Molecular Spectra and Molecular Structure. Iv. Constants of Diatomic Molecules; Van Nostrand Reinhold Co.: New York, 1979. 41. Khaenko, B. V.; Gnitetskii, O. A., Kristallografiya 1993, 38, 741. 42. Will, G.; Platzbecker, R., Z. anorg. allg. Chem 2001, 627, 2207. 43. Schultz; Ehrhart, Atomic Defects in Metals. Springer: Berlin, 1991; Vol. 25. 44. Korhonen, T.; Puska, M. J.; Nieminen, R. M., Vacancy-Formation Energies for Fcc and Bcc Transition Metals. Phys. Rev. B 1995, 51, 9526-9532. 45. Korzhavyi, P. A.; Abrikosov, I. A.; Johansson, B., First-Principles Calculations of the Vacancy Formation Energy in Transition and Noble Metals. Phys. Rev. B 1999, 59, 11693-11703. 46. Henkelman, G.; Arnaldsson, A.; Jonsson, H., A Fast and Robust Algorithm for Bader Decomposition of Charge Density. Comp. Mater. Sci. 2006, 36, 354-360. 47. Slater, J. C., Atomic Radii in Crystals. J. Chem. Phys. 1964, 41, 3199-3204. 48. Allred, A. L., Electronegativity Values from Thermochemical Data. J. Inorg. Nucl. Chem. 1961, 17, 215-221. 49. Pauling, L., The Nature of the Chemical Bond, 3 ed.; Cornell Univ., USA, 1960. 50. Cottrell, A. H.; Bilby, B. A., Dislocation Theory of Yielding and Strain Ageing of Iron. P. Phys. Soc. A 1949, 62, 49-62. 51. Metzner, H.; Sielemann, R.; Klaumtinzer, S.; Butt, R.; Semmler, W., Intrinsic Point Defects in Niobium: Trapping at 100pd and 111in Impurities Observed by Pac. Z. Phys. B 1985, 61, 267-281. 52. Derlet, P. M.; Nguyen-Manh, D.; Dudarev, S. L., Multiscale Modeling of Crowdion and Vacancy Defects in Body-Centered-Cubic Transition Metals. Phys. Rev. B 2007, 76, 054107. 53. Nguyen-Manh, D.; Horsfield, A. P.; Dudarev, S. L., Self-Interstitial Atom Defects in Bcc Transition Metals: Group-Specific Trends. Phys. Rev. B 2006, 73, 020101. 54. Li, C.-X.; Luo, H.-B.; Hu, Q.-M.; Yin, F.-X.; Umezawa, O.; Yang, R., Theoretical Investigations of Interstitial Atoms in Bcc Metals: Local Lattice Distortion and Diffusion Barrier. Comp. Mater. Sci. 2012, 58, 67-70. 55. Powers, R. W.; Doyle, M. V., Diffusion of Interstitial Solutes in the Group V Transition Metals. J. Appl. Phys. 1959, 30, 514-524. 56. Bokstein, B.; Razumovskii, I., Grain Boundary Diffusion and Segregation in Interstitial Solid Solutions Based on Bcc Transition Metals: Carbon in Niobium. Interface Sci. 2003, 11, 41-49. 57. Vasilenok, L. B.; Kablov, E. N.; Bokshtein, B. S.; Razumovskii, I. M., Diffusion of Carbon Along Grain Boundaries in Niobium. Dokl. Phys. 2000, 45, 588-591. 58. Wu, L.; Wang, Y.; Yan, Z.; Zhang, J.; Xiao, F.; Liao, B., The Phase Stability and Mechanical Properties of Nb–C System: Using First-Principles Calculations and Nano-Indentation. J. Alloy Compd. 2013, 561, 220–227. 59. Butler, W. H., Electron-Phonon Coupling in the Transition Metals: Electronic Aspects. Phys. Rev. B 1977, 15, 5267-5282. 60. Koch, C. C.; Scarbrough, J. O.; Kroeger, D. M., Effects of Interstitial Oxygen on the Superconductivity of Niobium. Phys. Rev. B 1974, 9, 888-897. 61. Dynes, R. C.; Varma, C. M., Superconductivity and Electronic Density of States. J. Phys. F: Met. Phys. 1976, 6, L215-L219.
28
62. Isaev, E. I.; Simak, S. I.; Abrikosov, I. A.; Ahuja, R.; K.Vekilov, Y.; Katsnelson, M. I.; Lichtenstein, A. I.; Johansson, B., Phonon Related Properties of Transition Metals, Their Carbides, and Nitrides: A First-Principles Study. J. Appl. Phys. 2007, 101, 123519.
29
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